34 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA In particular, the topology induced on the multiplicative group R* (resp. C*) of real (resp. complex) numbers ^0 by the topology of R (resp. C) makes R* (resp. C*) into a topological group. If G is a topological group (in multiplicative notation), the mappings (j^ y) i_> Xy~l and (x, y) H+ x~~ly are continuous mappings of G x G into G (3.11.5). For each ae G, the left and right translations x\-*ax and jci-» xa are homeomorphisms of G onto G, because they are bijective and continuous ((3.20.14) and (12.5)) and so are the inverse mappings xt-*aTlx and x\r+xa~l. For any a, b in G, the mapping xt-+axb (and in particular the inner auto- morphism xh+axa"1) is therefore a homeomorphism of G onto G (3.11.5). Since the mapping x\-^x"1 is bijective and equal to its inverse, it also is a homeomorphism of G onto G. (12.8.2) Let G be a topological group. (i) For each open (resp. closed) subset A of G, and each xeG, the sets jcA, AJC and A""1 (the set of all y~~1, where y e A) are open (resp. closed) in G. (ii) For each open set A in G and each subset B 0/G, the sets AB (the set of all products yz, where y e A and z e B) and BA are open in G. The assertions in (i) follow immediately from the preceding remarks, and (ii) follows from (i), since AB = (J Az is a union of open sets. zeB (12.8.2.1) On the other hand, if A and B are closed in G, it does not neces- sarily follow that AB is closed (cf. (12.10.5)). For example, if 9 is an irrational number, consider the subgroups Z and OZ of R. Then the subgroup Z + 0Z is not closed in R. To see why, observe that this subgroup is denumerable, and therefore is not the whole of R (2.2.17). Hence it is enough to prove the following result: (12.8.2.2) The only closed subgroups ofR are R itself and the subgroups of the form aZ, where a e R. Granted this, we cannot have both 1 = na and 9 = ma with m and n integers, because 9 is irrational, and the assertion of (12.8.2.1) then follows. To prove (12.8.2.2), we shall first show that a subgroup H of R is either discrete or dense in R. If H is not discrete, then for each s > 0 there exists x 7* 0 in the set H n [-e, +e]. Since the integer multiples nx (n e Z) belong to H, every interval of length greater than £ in R contains one of these points and therefore H is dense in R.rough f(a) and y, there is an endpoint u of ID belonging to the open